PHYSICAL, THERMAL,
AND MECHANICAL
PROPERTIES OF POLYMERS
A1.1 PHYSICAL PROPERTIES
Physical properties of polymers include molecular weight, molar volume, density,
degree of polymerization, crystallinity of material, and so on. Some of these are
discussed herewith in the following sections.
A1.1.1 Degree of Polymerization and Molecular Weight
First of all, let us discuss the degree of polymerization. The degree of polymerization
(DP)-n in a polymer molecule is defined as the number of repeating units in the polymer
chain. For example,
−(−CH𝟐 − CH𝟐 −)−n
The molecular weight of a polymer molecule is the product of the degree of polymerization and the molecular weight of the repeating unit. The polymer molecules are not
identical but are a mixture of many species with different degrees of polymerization, that
is, with different molecular weights. Therefore, in the case of polymers we talk about
the average values of molecular weights.
Biosurfaces: A Materials Science and Engineering Perspective, First Edition.
Edited by Kantesh Balani, Vivek Verma, Arvind Agarwal, Roger Narayan.
© 2015 The American Ceramic Society. Published 2015 by John Wiley & Sons, Inc.
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A1.1.1.1 Molecular Weight Averages. Suppose we have a set of values {x1 ,
x2, … , xn } and the corresponding probability of occurrence is given by {P1 , P2 , … , Pn },
then the average value is defined as follows:
∞
∑
Pi xi
i=0
A1.1.1.2 Number-Average Molecular Weight. If Ni is the number of
polymer molecules having the molecular weight Mi , then the “number-average”
probability of the given mass is given by:
N
Pi = ∑∞ i
j=0 Nj
The number-average molecular weight is given by:
[
]
∑∞
∞
∑
Ni
i=0 Mi Ni
Mi = ∑
Mn =
∑∞
∞
j=o Nj
j=0 Nj
i=o
The physical properties (such as transition temperature, viscosity, etc.) and mechanical properties (such as strength, stiffness, and toughness) depend on the molecular
weight of polymer. The lower the molecular weight, lower the transition temperature,
viscosity, and the mechanical properties. Due to increased entanglement of chains with
increased molecular weight, the polymer gets higher viscosity in molten state, which
makes the processing of polymer difficult.
A1.1.1.3 Weight-Average Molecular Weight. The weight-average probability is given by:
NM
Pi = ∑∞ i i
j=0 Nj Mj
The weight-average molecular weight is given by:
[
]
∑∞
∞
2
∑
Ni Mi
i=0 Ni Mi
Mi = ∑∞
Mw =
∑∞
j=0 Nj Mj
j=0 Nj Mj
i=0
A typical plot showing the number-average and weight-average molecular
weight is shown in Fig. A1.1. The number-average molecular weight is less than the
weight-average molecular weight (see Fig. A1.1). The degree of polymerization can be
calculated using the number-average molecular weight.
Degree of polymerization =
Number average molecular weight
Molecular weight of the repeat unit
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330
Amount/frequency
Number average
molecular weight
Weight average
molecular weight
Molecular weight
Figure A1.1. Average molecular weights of polymer.
A1.1.1.4 Polydispersity Index or Heterogeneity Index. The ratio of the
weight-average molecular weights to the number-average molecular weights is called
polydispersity index (PDI) or heterogeneity index, which measures the polydispersity of
the polymer mixture.
M
PDI = w
Mn
The dispersity measures heterogeneity of sizes of molecules or particles in the mixture. The mixture is called monodisperse if the molecules have the same size, shape, or
mass. If the molecules in the mixture have an inconsistent size, shape and mass distribution, the mixture is called polydisperse.
The natural polymers are generally monodisperse as all synthetic polymers are polydisperse with some exceptions. The PDI is equal to or greater than 1 where as the polymer
chains approach uniform chain length, the PDI tends to unity.
A1.1.2 Polymer Crystallinity: Crystalline and Amorphous Polymers
The polymeric chains being very large are found in the polymer in two forms as follows:
Lamellar crystalline form in which the chains fold and make lamellar structure
arranged in the regular manner and amorphous form in which the chains are in the irregular manner. The lamellae are embedded in the amorphous part and can communicate
with other lamellae via tie molecules (see Fig. A1.2). Polymer may be amorphous or
semi-crystalline in nature.
The %crystallinity is given by:
%Crystallinity =
𝜌c (𝜌s − 𝜌a )
× 100
𝜌s (𝜌c − 𝜌a )
𝜌c = density of the completely crystalline polymer,
𝜌a = density of the completely amorphous polymer,
𝜌s = density of the sample.
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Crystalline
lamellae
Amorphous
region
Tie molecule
Figure A1.2. Semi-crystalline polymer.
A typical range of crystallinity can be defined as amorphous (0%) to highly crystalline (>90%). The polymers having simple structural chains as linear chains and slow
cooling rate will result in good crystallinity as expected. In slow cooling, sufficient time
is available for crystallization to take place. Polymers having high degree of crystallinity
are rigid and have high melting point, but their impact resistance is low. However, amorphous polymers are soft and have lower melting points. For a solvent, it is important to
state that it can penetrate the amorphous part more easily than the crystalline part.
Examples of amorphous polymers: polystyrene and poly(methyl methacrylate).
Examples of crystalline polymers: polyethylene, and PET polyester.
Spherulites: if the molten polymer is cooled down, then the crystalline lamellae
grow in radial direction from a nucleus along the three dimensions leading to a spherical
structure called spherulite. The amorphous region is in between the crystalline lamellae
(Fig. A1.3). Spherulite formation and its diameter depend on various parameters such
as the number of nucleation sites, polymer molecule structure and rate of cooling. Due
to highly ordered lamellae in the spherulite, it shows higher density, hardness, tensile
Crystalline
lamellae
Tie molecule
Amorphous region
Figure A1.3. A typical structure of spherulite.
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strength, and Young’s modulus. The elasticity and impact resistance are shown, because
the lamellae are connected to amorphous regions.
A1.2 THERMAL PROPERTIES OF POLYMERS
In the amorphous region of the polymer, at lower temperature, the molecules of the polymer are in, say, frozen state, where the molecules can vibrate slightly but are not able
to move significantly. This state is referred as the glassy state. In this state, the polymer
is brittle, hard and rigid analogous to glass. Hence the name glassy state. The glassy
state is similar to a supercooled liquid where the molecular motion is in the frozen state.
The glassy state shows hard, rigid, and brittle nature analogous to a crystalline solid
with molecular disorder as a liquid. Now, when the polymer is heated, the polymer
chains are able to wiggle around each other, and the polymer becomes soft and flexible similar to rubber. This state is called the rubbery state. The temperature at which
the glassy state makes a transition to rubbery state is called the glass transition temperature Tg . Note that the glass transition occurs only in the amorphous region, and the
crystalline region remains unaffected during the glass transition in the semi-crystalline
polymer.
A1.2.1 Melting Point and Glass Transition Temperature
The glass transition temperature is the property of the amorphous region of the polymer,
whereas the crystalline region is characterized by the melting point. In thermodynamics, the transitions are described as first and second order transitions. Glass transition
temperature is the second order transition, whereas the melting point is the first order
transition (see Fig. A1.4). The value of glass transition temperature is not unique because
the glassy state is not in equilibrium. The value of glass transition temperature depends
on several factors such as molecular weight, measurement method, and the rate of heating or cooling.
Approximate values of glass transition temperatures of some polymers are listed in
Table A1.1.
The semi-crystalline polymer shows both the transitions corresponding to their crystalline and amorphous regions. Thus, the semi-crystalline polymers have true melting
temperatures (Tm ) at which the ordered phase turns to disordered phase, whereas the
amorphous regions soften over a temperature range known as the glass transition (Tg ).
It should be noted that amorphous polymers do not possess the melting point, but all
polymers possess the glass transition temperature.
The polymer melting point Tm is increased if the double bonds, aromatic groups,
bulky or large side groups are present in the polymer chain, because they restrict the
flexibility of the chain. The branching of chains causes the reduction of melting point,
as defects are produced because of the branching.
A1.2.1.1 Factors Affecting the Glass Transition Temperature. The
glass transition temperature depends on the mobility and flexibility (ease of the chain
segment to rotate along the chain backbone) of the polymeric chains. If the polymeric
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Second order
transition
Rubbery
state
Specific volume
Liquid
Glassy
state
Tg
First order
transition
Glass
Semicrystalline
solid
Melt state
Solid
crystalline
state
Crystalline
solid
Tm
Tg
Tm
Temperature
Figure A1.4. Melting point and glass transition temperature of polymer.
TAB L E A1.1. Glass Transition Temperatures of
Some Polymers
Polymer
Tg (∘ C)
Polytetrafluoroethylene
Polypropylene (isotactic)
Polystyrene
Poly(methylmethacrylate) (atactic)
Nylon 6,6
Polyethylene (LDPE)
Polyethylene (HDPE)
Polypropylene (atactic)
Polycarbonate
Poly(vinyl acetate) (PVAc)
Polyester(PET)
Poly(vinyl alcohol) (PVA)
Poly(vinyl chloride) (PVC)
−97
+100
+100
+105
+57
−120
−90
−18
+150
+28
+69
+85
+87
chains can move easily, then the glassy state can be converted to the rubbery state at
lower temperature, that is, the glass transition temperature is lower. If somehow the
mobility of the chains is restricted, then the glassy state is more stable, and it is difficult
to break the restriction causing the immobility of the polymer chains at the lower
temperature, because more energy is required to make the chains free. Thus, in this
case, the glass transition temperature is raised.
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I. Intermolecular Forces. Strong intermolecular forces cause higher Tg .
For example, PVC (Tg = 80 ∘ C) has stronger intermolecular forces than
polypropylene (Tg = −18 ∘ C) because of the dipole–dipole forces from the
C—Cl bond.
II. Chain Stiffness. The presence of the stiffening groups (such as amide, sulfone, carbonyl, p-phenylene etc.) in the polymer chain reduces the flexibility
of the chain, leading to higher glass transition temperature. For example,
polyethyleneterephthalete is stiffer than polyethylene adipate due to the
presence of benzene ring (see Fig. A1.5). Therefore, Tg value is higher for
polyethyleneterephthalate.
III. Cross-Linking. The cross-links between chains restrict rotational motion and
raise the glass transition temperature. Hence, higher cross-linked molecule will
show higher Tg than that with lower cross-linked molecule.
IV. Pendant groups. The presence of pendent group can change the glass transition
temperature.
(a) Bulky pendant groups: the presence of bulky pendant group, such as a benzene
ring, can restrict rotational freedom, leading to higher glass transition temperature. As in polystyrene, the presence of benzene ring increases the Tg (see
Fig. A1.6). In polypropylene, there is no benzene ring that leads to lower Tg
value (Fig. A1.6).
(b) Flexible pendant groups: the presence of flexible pendant groups, for
example, aliphatic chains, limits the packing of the chains and hence increases
O
O
C
C
O
CH2
CH2
O
Polyethyleneterephthalate,Tg=69 °C
O
O
CH2CH2
O
C
n
O
CH2CH2CH2CH2
C
n
Polyethylene adipate,Tg=−70 °C
Figure A1.5. Presence of benzene in polyethyleneterephthalete (top molecular chain) makes
it stiffer (hence higher Tg ) than polyethylene adipate (bottom molecular chain).
CH3
CH2
CH
CH2
CH
n
n
Atactic Polystyrene,Tg=100 °C Atactic Polypropylene,Tg=−18 °C
Figure A1.6. Role of bulky pendant groups in affecting glass transition temperature.
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CH3
CH2
C
CH3
CH2
n
C
n
COOCH3
COO(CH2)CH3
Poly methyl methacrylate, Tg=105 °C Poly butyl methacrylate, Tg=20 °C
Figure A1.7. Role of flexible pendant groups in affecting glass transition temperature.
the rotational motion, tending to less Tg value. In polybutylmethacrylate, the
presence of large aliphatic chain reduces the Tg value when compared with
that of polymethylmethacrylate (Fig. A1.7).
V. Plasticizers. Plasticizers are low molecular weight and non-volatile materials added to polymers to increase their chain flexibility. They reduce the
intermolecular cohesive forces between the polymer chains, which in turn
decrease Tg .
VI. Molecular Weight. The glass transition temperature is also affected by the
molecular weight of the polymer (Fig. A1.8). Tg is increased with the molecular
weight. The molecular weight is related to the glass transition temperature by
the Fox–Flory Equation:
Tg = Tg,∞ −
K
Mn
(Fox–Flory Equation)
Glass transition temperature (K)
where Tg,∞ is the glass transition temperature at the molecular weight of infinity,
and K is the empirical parameter called Fox–Flory parameter related to the free
volume inside the polymer. It is observed that Tg is increased up to the molecular weight of approximately 20 000, and after this limit, the Tg is not affected
appreciably.
20000
Molecular weight (g/mol)
Figure A1.8. Variation of glass transition temperature with molecular weight of polymer.
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A1.2.2 Mechanical Properties
It is of great importance to be familiar with some basic mechanical properties of the
material before its application in any field, such as how much it can be stretched, how
much it can be bent, how hard or soft it is, how it behaves on the application of repeated
load and so on.
a. Strength: In simple words, the strength is the stress required to break the sample. There are several types of the strength, namely tensile (stretching of the
polymer), compressional (compressing the polymer), flexural (bending of the
polymer), torsional (twisting of the polymer), impact (hammering) and so on. The
polymers follow the following order of increasing strength: linear < branched <
cross-linked < network.
Factors Affecting the Strength of Polymers
1. Molecular Weight: The tensile strength of the polymer rises with increase
in molecular weight and reaches the saturation level at some value of the
molecular weight (Fig. A1.9). The tensile strength is related to molecular
weight by the following equation.
𝜎=
𝜎∞ −
A
M
Strength (MPa)
𝜎 ∞ is the tensile strength of the polymer with molecular weight of
infinity. A is some constant, and M is the molecular weight. At lower
molecular weight, the polymer chains are loosely bonded by weak van
der Waals forces and the chains can move easily, responsible for low
strength, although crystallinity is present. In case of large molecular weight
polymer, the chains become large and hence are entangled, giving strength
to the polymer.
2. Cross-linking: The cross-linking restricts the motion of the chains and
increases the strength of the polymer.
Molecular weight (g/mol)
Figure A1.9. Variation of tensile strength with molecular weight of the polymer.
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3. Crystallinity: The crystallinity of the polymer increases strength, because
in the crystalline phase, the intermolecular bonding is more significant.
Hence, the polymer deformation can result in the higher strength leading
to oriented chains.
b. Percent Elongation to Break (Ultimate Elongation): It is the strain in the material
on its breakage, as shown in Fig. A1.10. It measures the percentage change in the
length of the material before fracture. It is a measure of ductility. Ceramics have
very low (<1%), metals have moderate (1–50%) and thermoplastic (>100%),
thermosets (<5%) value of elongation to break.
c. Young’s Modulus (Modulus of Elasticity or Tensile Modulus): Young’s Modulus
is the ratio of stress to the strain in the linearly elastic region (Fig. A1.11). Elastic
modulus is a measure of the stiffness of the material.
E=
Tensile Stress(𝜎)
Tensile Strain(𝜀)
d. Toughness: The toughness of a material is given by the area under a stress–strain
curve (Fig. A1.12).
Stress (MPa)
Toughness =
∫
𝜎 d𝜀
Break
Elongation
Stress (MPa)
Figure A1.10. Elongation to break of the polymer.
Slope=E
Strain
Figure A1.11. Young’s modulus of the polymer.
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Stress (MPa)
Break
Toughness
Strain
Figure A1.12. The toughness of polymer material.
Stress (MPa)
Brittle polymer
(Glassy polymer/low temperature thermoset)
Ductile polymer
(semi crystalline polymer
/plastic/high temperature
thermoplastic)
Highly elastic
(elastomer)
Strain
Figure A1.13. Stress–strain behavior of different types of materials.
The toughness measures the energy absorbed by the material before it breaks.
A typical stress–strain curve is shown in Fig. A1.13, which compares the
stress–strain behavior of different types of materials. The rigid materials possess high Young’s modulus (such as brittle polymers), and ductile polymers also
possess similar elastic modulus, but with higher fracture toughness. However,
elastomers have low values of Young’s modulus and are rubbery in nature.
The yield strength of the plastic polymer is the corresponding stress where
the elastic region (linear portion of the curve) ends (Fig. A1.14). The tensile
strength is the stress corresponding to the fracture of the polymer. The tensile
strength may be higher or lower than the yield strength (Fig. A1.14).
The mechanical properties of the polymer are strongly affected by the temperature. A typical plot of stress versus strain is shown in Fig. A1.15. From the
plot, it is clear that with increase in the temperature, the elastic modulus and
tensile strength are decreased, but the ductility is enhanced.
e. Viscoelasticity: There are two types of deformations: elastic and viscous. Consider the constant stress level applied to a material as shown in the Fig. A1.16.
In the elastic deformation (Fig. A1.17), the strain is generated at the moment
the constant load (or stress) is applied, and this strain is maintained until the stress
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Stress
T2
T3
Ductile polymer
T4
Stress (MPa)
Stress (MPa)
Fracture
Yield
strength
Strain
Figure A1.14. Yield strength and tensile strength of polymer.
Tensile
strength
Brittle polymer
T1
Time
tr
ta
Temperature:
T1<T2<T3<T4
Strain
Figure A1.15. Effect of temperature on the mechanical properties of polymer.
σo
Figure A1.16. Constant stress applied to a polymer.
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Strain
εo
ta
Time
tr
Figure A1.17. Elastic deformation.
is not released. On removal of the stress, the material recovers its original dimensions completely, that is the deformation is reversible (Fig. A1.17), that is:
𝜎 = E𝜀
where E is the elastic modulus, 𝜎 is applied stress, and 𝜖 is the strain developed.
However, in viscous deformation (Fig. A1.18), the strain generated is not instantaneous and it is time dependent. The strain keeps on increasing with time on application of
the constant load, that is, the recovery process is delayed. When the load is removed, the
material does not return to its original dimensions completely, that is, this deformation
is irreversible (Fig. A1.18).
d𝜀
𝜎=𝛾
dt
where
Strain
𝛾 = viscosity, and
d𝜖/dt = strain rate
ta
Time
tr
Figure A1.18. Viscous deformation.
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ta
tr
Viscous
behavior
ta
Time
Strain
Strain
Strain
Elastic
behavior
Time
Viscoelastic
behavior
ta
tr
tr
Time
Figure A1.19. Viscoelastic deformation: the combined behavior of viscosity and elasticity.
Usually, polymers show a combined behavior of elastic and plastic deformation
(Fig. A1.19) depending on the temperature and strain rate. At low temperature and high
strain rate, elastic behavior is observed, and at high temperature but low strain rate,
the viscous behavior is observed. The combined behavior of viscosity and elasticity is
observed at intermediate temperature and strain rate values. This behavior is termed as
viscoelasticity, and the polymer is termed as viscoelastic.
A1.2.2.1 Viscoelastic Relaxation Modulus. At a given temperature, when
the polymer is strained to a given value, then the stress required to maintain this
strain is found to decrease with time. This is called stress relaxation (Fig. A1.20). The
stress required to maintain the constant strain value is decreased with time, because
the molecules of polymer get relaxed with time, and to maintain the level of strain,
somewhat lower value of stress is sufficient (Fig. A1.20).
𝜎(t)
𝜀0
Strain
Erel (t) =
εo
Time
tο
Time
Stress
tο
σo
Figure A1.20. Stress relaxation in polymer.
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Log erel(t)
Temperature:
T1< T2<T3< T4<T5
T1
T2
T3
T4
T5
t1
Log t
Figure A1.21. Variation of relaxation modulus with temperature and time.
Relaxation modulus erel(t1)
Glassy
Glass transition region
/leathery
Rubbery
Rubbery flow
Viscous flow
Tg:
Glass transition temperature
Tm:
Melting point
Temperature T
Figure A1.22. Variation of relaxation modulus with temperature after a given time t.
The decrease in stress follows the exponential decay:
𝝈 = 𝝈 o e−t∕𝝉
where
𝜎 = stress at time t,
𝜎 o = peak stress level, and
𝜏 = relaxation time constant.
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Stress
σο
Time
Strain
tο
εο
tο
Time
Figure A1.23. Viscoelastic creep.
The relaxation modulus is found to decrease with increase in temperature and time
as shown in Fig. A1.21.
Now, consider the time t1. Measure the values of relaxation modulus at time t1
at different temperatures, say T1, T2, T3 … for the viscoelastic polymer and plot the
relaxation modulus versus temperature. A typical plot is shown in Fig. A1.22. The glass
transition temperature lies near the upper temperature extremity.
Viscoelastic Creep: creep can be considered as the opposite of stress relaxation
(lowering of the stress with time maintaining the constant strain level) where the polymer suffers time-dependent deformation (increasing strain with time) at constant stress
level. At a given temperature, when a constant load is applied to the material, there is
a time-dependent increase of strain in the material (Fig. A1.23). This behavior of the
material is called viscoelastic creep. The increase in molecular weight and stiffness of
the chains results in better creep resistance of the material.
We can define the time-dependent creep modulus as follows:
ECreep (t) =
𝜎0
𝛆(t)
Or we can define the creep compliance as follows:
JCreep (t) =
where
𝜎 o = constant applied stress,
𝜖(t) = strain at time t.
𝛆(t)
𝜎0
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